In the orthonormal frame, the comoving <perfect fluid in general relativity> has $T_{00}=\rho$ and $T_{ij}=p\delta_{ij}$. The <Einstein field equations> with <cosmological constant> therefore give
$$
\frac34-\Lambda=8\pi\rho,
\qquad
-\frac14+\Lambda=8\pi p.
$$
Thus
$$
\boxed{\rho=\frac{3/4-\Lambda}{8\pi},
\qquad p=\frac{\Lambda-1/4}{8\pi}}.
$$
Choosing
$$
\boxed{\Lambda=\frac14}
$$
gives $p=0$ and $\rho=1/(16\pi)>0$. The metric is then the <Einstein static universe> supported by pressureless matter and positive vacuum energy.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 4 / d / Solution

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The Einstein static universe is a product of time with a round three-sphere supported by matter and a positive cosmological constant. For pressureless matter its cosmological repulsion balances the spatial curvature exactly.


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